Fourier and wavelet transforms have proven to be indispensable tools in signal processing. They are taught in many courses, both at the graduate and undergraduate levels. When deciding how to teach these topics, the lecturer can choose from a number of different paths. Goodman’s textbook is for an undergraduate course based on an approach that is strongly focused on the relation between the two types of transforms on the one hand and the field of linear algebra on the other. While introducing Fourier and wavelet transforms in this manner, the book simultaneously provides a gentle introduction to MATLAB.

The core of the book can be subdivided into three parts. The first part is chapter 1, an introduction to the basic elements of signal processing (such as sampling and quantization) and to the fundamental concepts of linear algebra that are needed later in the book. This chapter is written on a level suitable for beginners. Therefore, the book is accessible to students from a variety of different areas; a strong mathematical background is not required.

The second part of the book consists of chapter 2 and deals with the discrete Fourier transform in general and the fast Fourier transform in particular. As already indicated, Goodman takes a strong linear algebra standpoint and develops the presentation based on the concept of matrices. Consequently, analytical aspects of the topic are not discussed very deeply.

The third and longest--by a large amount--part of the book comprises chapters 3, 4, and 5. This part is devoted to wavelets and their application in signal processing. It starts by introducing Haar wavelets and by using them to demonstrate the basic idea of the wavelet transform. Other types of wavelets up to the Daubechies4 concept are briefly mentioned here as well in preparation of a more detailed discussion in chapter 4. However, before that point is reached, Goodman continues chapter 3 with an exploration of lifting schemes, of multi-resolution analysis, and of wavelet bases and their use for single- and multi-scale transforms in one and two dimensions. Chapter 4 then begins with the transfer of the classical filter concept to the world of wavelets, and continues with a description of filter banks and their possible applications. While these two chapters are essentially focused on discrete wavelet transforms, the application of wavelets to continuous signals is addressed in chapter 5.

Each chapter begins with an overview section that tells the reader what to expect in the chapter; sometimes the most fundamental notions and concepts are briefly mentioned here as well. I think this is a very useful feature, in particular for inexperienced readers like students wanting to use the book for self-study, because it raises awareness for certain connections between different aspects and thus helps with understanding how things are interrelated. Moreover, each chapter ends with a section entitled “Computer Explorations” where the reader is guided through a number of small MATLAB programs that elucidate what was taught in the respective chapter, thus deepening the reader’s understanding and at the same time teaching the use of the MATLAB package. Plenty of exercises are provided for each chapter as well; the corresponding solutions are listed in an appendix. Another appendix provides a few elementary prerequisites like a brief introduction to complex numbers and the exponential function, and a few words about MATLAB and the Uvi_Wave collection of MATLAB files that is available from the book’s companion website and that the reader can use to perform some signal processing transforms in practice.

The book’s structure follows a carefully designed concept. It is well written and readable. The only weaknesses are the lack of a discussion of the concept of frames and the fact that the subject index is organized in a very unclear and confusing way that makes it very difficult to find particular keywords. In spite of these relatively minor deficiencies, Goodman’s book can be recommended as a starting point for developing a beginner’s course on wavelets or for self-study of the field. It is clearly of an introductory nature and does not provide the deepest and most advanced results, but it is definitely sufficient to obtain an overview of what can be done with Fourier and wavelet transforms in signal processing, particularly with respect to the linear algebra-related aspects of the field. The accompanying MATLAB source code files allow readers to easily compute such transforms for themselves.